Question: Solve the equation. $\dfrac{dy}{dx}=-\dfrac{\sin(x+5)}{y}$ Choose 1 answer: Choose 1 answer: (Choice A) A $y=\pm\sqrt{2\cos(x+5)}+C$ (Choice B) B $y=\pm\sqrt{2\cos(x+5)+C}$ (Choice C) C $y=Ce^{-\cos(x+5)}$ (Choice D) D $y=\pm e^{-\cos(x+5)}+C$
Explanation: We can bring this equation to the form $f(y)\,dy=g(x)\,dx$ : $\begin{aligned} \dfrac{dy}{dx}&=-\dfrac{\sin(x+5)}{y} \\\\ -y\,dy&=\sin(x+5)\,dx \end{aligned}$ This means we can solve this equation using separation of variables! $\begin{aligned} -y\,dy&=\sin(x+5)\,dx \\\\ \int -y\,dy&=\int \sin(x+5)\,dx \\\\ -\dfrac{y^2}{2}&=-\cos(x+5)+C_1 \\\\ y^2&=2\cos(x+5)+C \\\\ y&=\pm\sqrt{2\cos(x+5)+C} \end{aligned}$ [Where did we get C?] Notice that after the integration, more work was required in order to isolate $y$. In conclusion, this is the solution of the equation: $y=\pm\sqrt{2\cos(x+5)+C}$